Let $B$ be a finite dimensional selfinjective algebra over a field $k$ with a finite dimensional non-projective $B$-module $M$ and $$A=\pmatrix{k&M\\0&B}.$$

A module $N$ over an algebra $C$ is called Gorenstein projective in case $Ext_C^i(N,C)=0=Ext_C^i(D(C),\tau(N))$ for all $i >0$.

Questions:

Is there a general description of Gorenstein projective $A$-modules depending on $B$ and $M$?

Can $A$ have only finitely many indecomposable Gorenstein projectives when $M$ satisfies $Ext_B^j(M,M)=0$ for all $j >t$ for some $t$?

Together with (the answer in) Question on Ext for finite dimensional algebras a positive answer to 2. would give a negative answer to the first question in chapter 8 of https://arxiv.org/pdf/1808.01809.pdf.

There is some literature on triangular matrix algebras and their Gorenstein projective modules, but it seems that the assumptions are always too strong to apply here.